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2ndle PDF

1. Find the electric potential at a point P far from the origin due to 4 charged particles placed at specific locations, expressing the potential up to the quadrupole term using spherical coordinates. 2. Find the electric field and charge distribution for a system of two concentric conducting spheres with opposite charges, where the space between is half filled with a dielectric hemisphere. 3. Find the electric field and electric displacement inside and outside a dielectric sphere with a polarized charge distribution of Kr-2.

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365 views1 page

2ndle PDF

1. Find the electric potential at a point P far from the origin due to 4 charged particles placed at specific locations, expressing the potential up to the quadrupole term using spherical coordinates. 2. Find the electric field and charge distribution for a system of two concentric conducting spheres with opposite charges, where the space between is half filled with a dielectric hemisphere. 3. Find the electric field and electric displacement inside and outside a dielectric sphere with a polarized charge distribution of Kr-2.

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docx
Copyright
© © All Rights Reserved
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PHY605

EXAM 2
(Due Saturday, 1 December)

(Hardcopies only, handwritten on only one side of the paper.)


z
1. Four particles, (one of charge q, one of charge 3q, and two of
• 3q
charge −2q) are placed as shown in the figure, each a distance d
from the origin. Find the electric potential at a point P far from the
origin, in terms of spherical coordinates (r, θ, φ), up to the • •
quadrupole term. −2q −2q y

x • q

2. Two concentric conducting spheres of inner and outer radii a and b,


respectively, carry charges ±Q. The empty space between the spheres is
half-filled by a hemispherical shell of dielectric (of permittivity 𝜖), as shown
in the figure.
a) Find the electric field everywhere between the spheres.
b) Calculate the charge distribution on the surface of the inner sphere.
c) Calculate the polarization-charge density induced on the surface of
the dielectric at r = a.

3. A dielectric sphere of radius a has a polarization 𝐏 = 𝐾𝑟 ! 𝐫 . Find the electric field and electric
displacement at distance 𝑟 from center, a) for 𝑟 < 𝑎 (inside the sphere), and b) for 𝑟 > 𝑎 (outside the
sphere)

z
4. A point charge q is embedded in a dielectric of
permittivity 𝜖 , at distance 𝑑 below the surface, as O
shown in the figure. The surface of the dielectric is at x
𝑧 = 0. d
a) Find the electrostatic potential at points in
(𝑧 < 0) and outside (𝑧 > 0) the dielectric. q
b) Find the polarization charge density on the
surface (𝑧 = 0) of the dielectric.

5. A thick spherical shell of inner radius a and outer radius b is made of dielectric
!
material with a “frozen-in” polarization 𝐏 𝐫 = 𝐫, where k is a constant and r is
! b
the distance from the center. (There is no free charge in the problem.) Find the
electric field in all three regions (𝑟 < 𝑎, 𝑎 < 𝑟 < 𝑏, and 𝑟 > 𝑎) by two different
methods:
a
(a) Locate all the bound charge, and use Gauss’s law to calculate the field it
produces.
(b) Find D and solve for E.

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